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author | Prefetch | 2022-10-20 18:25:31 +0200 |
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committer | Prefetch | 2022-10-20 18:25:31 +0200 |
commit | 16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch) | |
tree | 76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/spherical-coordinates/index.md | |
parent | e5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff) |
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/spherical-coordinates/index.md')
-rw-r--r-- | source/know/concept/spherical-coordinates/index.md | 34 |
1 files changed, 17 insertions, 17 deletions
diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md index bc01c95..9df5d51 100644 --- a/source/know/concept/spherical-coordinates/index.md +++ b/source/know/concept/spherical-coordinates/index.md @@ -10,16 +10,16 @@ layout: "concept" **Spherical coordinates** are an extension of polar coordinates to 3D. The position of a given point in space is described by -three coordinates $(r, \theta, \varphi)$, defined as: +three coordinates $$(r, \theta, \varphi)$$, defined as: -* $r$: the **radius** or **radial distance**: distance to the origin. -* $\theta$: the **elevation**, **polar angle** or **colatitude**: - angle to the positive $z$-axis, or **zenith**, i.e. the "north pole". -* $\varphi$: the **azimuth**, **azimuthal angle** or **longitude**: - angle from the positive $x$-axis, typically in the counter-clockwise sense. +* $$r$$: the **radius** or **radial distance**: distance to the origin. +* $$\theta$$: the **elevation**, **polar angle** or **colatitude**: + angle to the positive $$z$$-axis, or **zenith**, i.e. the "north pole". +* $$\varphi$$: the **azimuth**, **azimuthal angle** or **longitude**: + angle from the positive $$x$$-axis, typically in the counter-clockwise sense. -Cartesian coordinates $(x, y, z)$ and the spherical system -$(r, \theta, \varphi)$ are related by: +Cartesian coordinates $$(x, y, z)$$ and the spherical system +$$(r, \theta, \varphi)$$ are related by: $$\begin{aligned} \boxed{ @@ -31,8 +31,8 @@ $$\begin{aligned} } \end{aligned}$$ -Conversely, a point given in $(x, y, z)$ -can be converted to $(r, \theta, \varphi)$ +Conversely, a point given in $$(x, y, z)$$ +can be converted to $$(r, \theta, \varphi)$$ using these formulae: $$\begin{aligned} @@ -47,7 +47,7 @@ $$\begin{aligned} The spherical coordinate system is an orthogonal [curvilinear system](/know/concept/curvilinear-coordinates/), -whose scale factors $h_r$, $h_\theta$ and $h_\varphi$ we want to find. +whose scale factors $$h_r$$, $$h_\theta$$ and $$h_\varphi$$ we want to find. To do so, we calculate the differentials of the Cartesian coordinates: $$\begin{aligned} @@ -58,7 +58,7 @@ $$\begin{aligned} \dd{z} &= \dd{r} \cos\theta - \dd{\theta} r \sin\theta \end{aligned}$$ -And then we calculate the line element $\dd{\ell}^2$, +And then we calculate the line element $$\dd{\ell}^2$$, skipping many terms thanks to orthogonality: $$\begin{aligned} @@ -74,7 +74,7 @@ $$\begin{aligned} Finally, we can simply read off the squares of the desired scale factors -$h_r^2$, $h_\theta^2$ and $h_\varphi^2$: +$$h_r^2$$, $$h_\theta^2$$ and $$h_\varphi^2$$: $$\begin{aligned} \boxed{ @@ -146,7 +146,7 @@ $$\begin{aligned} } \end{aligned}$$ -The differential element of volume $\dd{V}$ +The differential element of volume $$\dd{V}$$ takes the following form: $$\begin{aligned} @@ -163,7 +163,7 @@ $$\begin{aligned} = \int_0^{2\pi} \int_0^\pi \int_0^\infty f(r, \theta, \varphi) \: r^2 \sin\theta \dd{r} \dd{\theta} \dd{\varphi} \end{aligned}$$ -The isosurface elements are as follows, where $S_r$ is a surface at constant $r$, etc.: +The isosurface elements are as follows, where $$S_r$$ is a surface at constant $$r$$, etc.: $$\begin{aligned} \boxed{ @@ -177,7 +177,7 @@ $$\begin{aligned} } \end{aligned}$$ -Similarly, the normal vector element $\dd{\vu{S}}$ for an arbitrary surface is given by: +Similarly, the normal vector element $$\dd{\vu{S}}$$ for an arbitrary surface is given by: $$\begin{aligned} \boxed{ @@ -188,7 +188,7 @@ $$\begin{aligned} } \end{aligned}$$ -And finally, the tangent vector element $\dd{\vu{\ell}}$ of a given curve is as follows: +And finally, the tangent vector element $$\dd{\vu{\ell}}$$ of a given curve is as follows: $$\begin{aligned} \boxed{ |